Upgrading and Downgrading Torus Actions

TL;DR

This paper explores methods to modify p-divisors to describe affine T-varieties under larger tori or subtori, providing explicit constructions and applications to Cox rings and deformations of toric varieties.

Contribution

It introduces techniques for upgrading and downgrading p-divisors to handle different torus actions on affine T-varieties, expanding their descriptive framework.

Findings

Explicit constructions for p-divisors of Cox rings.

Methods to upgrade p-divisors for larger tori.

Approaches to downgrade p-divisors for subtori.

Abstract

K. Altmann and J. Hausen have shown that affine T-varieties can be described in terms of p-divisors. Given a p-divisor describing a T-variety X, we show how to construct new p-divisors describing X with respect to actions by larger tori. Conversely, if dim T=dim X-1, we show how to construct new p-divisors describing X with respect to actions by closed subtori of T. As a first application, we give explicit constructions for the p-divisors describing certain Cox rings. Furthermore, we show how to upgrade the p-divisors describing the total spaces of homogeneous deformations of toric varieties.